SecondOrderODEs - Maple Help

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ODE Steps for Second Order ODEs

 

Overview

Examples

Overview

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This help page gives a few examples of using the command ODESteps to solve second order ordinary differential equations.

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See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

withStudent:-ODEs:

ode12xdiffyx,x9x2+2diffyx,x+x2+1diffyx,x,x=0

ode12xⅆⅆxyx9x2+2ⅆⅆxyx+x2+1ⅆ2ⅆx2yx=0

(1)

ODEStepsode1

Let's solve2xⅆⅆxyx9x2+2ⅆⅆxyx+x2+1ⅆ2ⅆx2yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxMake substitutionu=ⅆⅆxyxto reduce order of ODE2xux9x2+2ux+x2+1ⅆⅆxux=0Check if ODE is exactODE is exact if the lhs is the total derivative of aC2function=0Compute derivative of lhsxFx,u+uFx,uⅆⅆxux=0Evaluate derivatives2x=2xCondition met, ODE is exactExact ODE implies solution will be of this formFx,u=_C1,Mx,u=xFx,u,Nx,u=uFx,uSolve forFx,uby integratingMx,uwith respect toxFx,u=+_F1uEvaluate integralFx,u=x2u3x3+_F1uTake derivative ofFx,uwith respect touNx,u=uFx,uCompute derivativex2+2u+1=x2+ⅆⅆu_F1uIsolate forⅆⅆu_F1uⅆⅆu_F1u=2u+1Solve for_F1u_F1u=u2+uSubstitute_F1uinto equation forFx,uFx,u=x2u3x3+u2+uSubstituteFx,uinto the solution of the ODEx2u3x3+u2+u=_C1Solve foruxux=x2212x4+12x3+2x2+4_C1+12,ux=x2212+x4+12x3+2x2+4_C1+12Solve 1st ODE foruxux=x2212x4+12x3+2x2+4_C1+12Make substitutionu=ⅆⅆxyxⅆⅆxyx=x2212x4+12x3+2x2+4_C1+12Integrate both sides to solve foryxⅆⅆxyxⅆx=x2212x4+12x3+2x2+4_C1+12ⅆx+_C2Compute lhsyx=x2212x4+12x3+2x2+4_C1+12ⅆx+_C2Solve 2nd ODE foruxux=x2212+x4+12x3+2x2+4_C1+12Make substitutionu=ⅆⅆxyxⅆⅆxyx=x2212+x4+12x3+2x2+4_C1+12Integrate both sides to solve foryxⅆⅆxyxⅆx=x2212+x4+12x3+2x2+4_C1+12ⅆx+_C2Compute lhsyx=x2212+x4+12x3+2x2+4_C1+12ⅆx+_C2

(2)

ode2diffyx,x,xdiffyx,xxexpx=0

ode2ⅆ2ⅆx2yxⅆⅆxyxxⅇx=0

(3)

ODEStepsode2

Let's solveⅆ2ⅆx2yxⅆⅆxyxxⅇx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=ⅆⅆxyx+xⅇxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yxⅆⅆxyx=xⅇxCharacteristic polynomial of homogeneous ODEr2r=0Factor the characteristic polynomialrr1=0Roots of the characteristic polynomialr=0,11st solution of the homogeneous ODEy1x=12nd solution of the homogeneous ODEy2x=ⅇxGeneral solution of the ODEyx=_C1y1x+_C2y2x+ypxSubstitute in solutions of the homogeneous ODEyx=_C1+_C2ⅇx+ypxFind a particular solutionypxof the ODEUse variation of paramaters to findypherefxis the forcing functionypx=y1xy2xfxWy1x,y2xⅆx+y2xy1xfxWy1x,y2xⅆx,fx=xⅇxWronskian of solutions of the homogeneous equationWy1x,y2x=1ⅇx0ⅇxCompute WronskianWy1x,y2x=ⅇxSubstitute functions into equation forypxypx=xⅇxⅆx+ⅇxxⅆxCompute integralsypx=ⅇxx22x+22Substitute particular solution into general solution to ODEyx=_C1+_C2ⅇx+ⅇxx22x+22

(4)

ode3diffyx,x,x+5diffyx,x2yx=0

ode3ⅆ2ⅆx2yx+5ⅆⅆxyx2yx=0

(5)

ODEStepsode3

Let's solveⅆ2ⅆx2yx+5ⅆⅆxyx2yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxDefine new dependent variableyux=Computeⅆ2ⅆx2yx=Use chain rule on the lhs=Substitute in the definition ofuuy=Make substitutionsⅆⅆxyx=uy